The book is an account on recent advances in elliptic and parabolic problems and related equations, including general quasi-linear equations, variational structures, Bose-Einstein condensate, Chern-Simons model, geometric shell theory and stability in fluids. It presents very up-to-date research on central issues of these problems such as maximal regularity, bubbling, blowing-up, bifurcation of solutions and wave interaction. The contributors are well known leading mathematicians and prominent young researchers.
Author(s): Chiun-Chuan Chen, Michel Chipot, Chang-Shou Lin
Publisher: World Scientific Publishing Company
Year: 2005
Language: English
Pages: 283
CONTENTS......Page 6
Preface......Page 4
1. Introduction......Page 8
2. Abstract theory......Page 9
3. Parabolic boundary value problems: weak settings......Page 11
4. Model problems......Page 15
5. Parabolic boundary value problems: strong settings......Page 19
References......Page 22
Assis Azevedo, Jose-Francisco Rodrigues and Lisa Santos Remarks on the Two and Three Membranes Problem......Page 26
1. Introduction and notations......Page 27
2. The N = 2 membranes problem......Page 30
3. The N = 3 membranes problem......Page 34
Acknowledgment......Page 39
References......Page 40
1. Introduction......Page 42
2. The Proofs of Theorem 1.1......Page 44
References......Page 45
1.1. Blowing-up solutions in Problem (1)......Page 48
1.2. Blowing-up solutions in Problem (2)......Page 50
1.3. Multiple bubbling at a single point......Page 52
2. Sketch of Proof of Theorem 1.1......Page 54
3. Construction of blowing-up solutions for Problem (2)......Page 62
References......Page 64
2. A physical model......Page 68
3. A "model" problem......Page 70
4. Proof of Theorem 3.1......Page 74
5. Existence of blowing up solutions......Page 78
References......Page 84
Jacques Giacomoni, Jyotshana Prajapat and Mythily Ramaswamy Positive Solution Branch for Elliptic Problems with Critical Indefinite Nonlinearity......Page 88
1. Introduction......Page 89
2. An outline of the proof......Page 91
3. Estimates in s......Page 92
5. Estimates in ......Page 93
5.1. Blow up points of {vi}......Page 94
5.2. Nature of Blow up points of vi......Page 97
6. Solution branch......Page 99
References......Page 101
1. Introduction. The classical Marguerre-von Karman equations......Page 104
2. The generalized Marguerre-von Karman equations as a more general semi-linear elliptic system......Page 106
3. Existence theory......Page 111
Acknowledgment......Page 112
References......Page 113
2. Generalized Fundamental Solutions......Page 114
2.1. Some Definitions and Notations......Page 115
3.1. Elliptic Regularizations of Parabolic Problems......Page 118
3.2. Abstract Elliptic Problems......Page 121
3.3. A Hyperbolic Problem......Page 123
References......Page 126
1. Introduction......Page 128
2. Existence for the case c > 0......Page 130
3. Partial uniqueness and uniqueness for the case c > 0......Page 134
4. The balanced bistable case......Page 139
References......Page 141
1. Introduction......Page 144
2. Abelian Higgs Model......Page 145
3. Chern-Simons-Higgs Model......Page 148
4. Maxwell-Chern-Simons-Higgs Model......Page 150
References......Page 156
1. Introduction......Page 160
2. Proof of Theorem 1.1......Page 163
2.1. Set up the problem......Page 164
2.2. Solutions to the limit equation as e 0......Page 165
2.3. Solutions for e > 0......Page 168
2.4. Applying the implicit function theorem......Page 169
3. Asymptotic behavior......Page 171
3.1. Proof of Theorem 1.2......Page 172
3.2. Proof of Lemma 3.1.......Page 175
3.4. Proof of Lemma 3.2.......Page 178
3.5. Proof of Lemma 3.3.......Page 180
References......Page 184
1. Introduction......Page 186
2. Doubling Inequaltiy......Page 189
1. Introduction and the Main Results......Page 196
2. Computations and Figures......Page 201
References......Page 203
1. Introduction......Page 204
2. The result......Page 205
3. Sketch of the proof......Page 206
References......Page 207
1. Introduction......Page 210
2. Position of the problem......Page 211
Conservation and balance laws......Page 212
Results on Equilibrium Shapes......Page 213
3. Formal tools......Page 215
First and Second Variation of......Page 216
4. Free work identity......Page 217
5. Proof of nonlinear instability......Page 219
References......Page 222
1. Introduction......Page 224
2. An explicit example of incomplete blow-up......Page 227
3. Complete blow-up for non-autonomous problems......Page 228
4. Indefinite problems......Page 230
5. Problems with nonlinear boundary conditions......Page 232
6. Proof of Theorem 3.2......Page 233
References......Page 235
1. Introduction......Page 238
2. Auxiliary propositions......Page 245
3. Proof of Theorem 1.1......Page 254
References......Page 264
1. Introduction......Page 266
2. Preliminaries......Page 269
3. Perturbation theorem of the Fredholm operator......Page 272
References......Page 276
1. Introduction......Page 278
References......Page 283
